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Math Help - Quick question about the empty set and the power set

  1. #1
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    Quick question about the empty set and the power set

    Let's say we have a set A = {1, 2, 3, 4}

    I understand that, for example, \left\{{2, 3}\right\}\in P(A), but why is it that \left\{{\varnothing}\right\}\subseteq P(A)?
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  2. #2
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    Re: Quick question about the empty set and the power set

    Take a subset S of A.
    Then this subset is an element of P(A).
    S is in P(A)
    For a set to be a subset of P(A), each of its elements must be in P(A).
    Ie each of its elements must be a subset of A.
    The empty set is a subset of A.
    The set containing the empty set is therefore a subset of P(A)
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  3. #3
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    Re: Quick question about the empty set and the power set

    So if I took the set {{2, 3}} it becomes a subset of P(A) rather than an element, correct? I think I understand, thank you.
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    Re: Quick question about the empty set and the power set

    Quote Originally Posted by Robinator View Post
    Let's say we have a set A = {1, 2, 3, 4}

    I understand that, for example, \left\{{2, 3}\right\}\in P(A), but why is it that \left\{{\varnothing}\right\}\subseteq P(A)?
    Is every member of {0} a member of PA? Yes, since 0 is the only member of {0} while 0 is in PA. So {0} is a subset of PA.
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  5. #5
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    Re: Quick question about the empty set and the power set

    Quote Originally Posted by Robinator View Post
    So if I took the set {{2, 3}} it becomes a subset of P(A) rather than an element, correct?
    Yes, {{2 3}} is a subset of PA, and {{2 3}} is a member of PPA, but {{2 3}} is not a member of PA.
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